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### Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
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### Given $x+y$ and $x\cdot y$, what is $x^3+ y^3$ ?

I have been looking at an assortment of high school number sense tests and I noticed a reoccurring problem that states what x+y is and what $x\cdot y$ is then asks for $x^3+ y^3$. I want to know how ...
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### Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?

Are there any computer algebra systems with the functionality to allow me to enter in an explicit symmetric polynomial and have it return that polynomial in terms of the elementary symmetric ...
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### If $x + \frac{1}{x} = \sqrt{3}$, then find $x^{18}$ [duplicate]

If $x + \frac{1}{x} = \sqrt{3}$, then find $x^{18}$. This is sorta like If $x^3+\frac{1}{x^3}=18\sqrt{3}$ then to prove $x=\sqrt{3}+\sqrt{2}$. So my question is, do I solve my question the same way as ...